C2^4:2Dic5 - GroupNames

G = C₂⁴⋊₂Dic₅ order 320 = 2⁶·5

1^st semidirect product of C₂⁴ and Dic₅ acting via Dic₅/C₅=C₄

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴⋊₂Dic₅, C₅⋊₅C₂≀C₄, (C₂×C₂₀).₃D₄, (C₂³×C₁₀)⋊₃C₄, (C₂×D₄).₇D₁₀, C₂₀.D₄⋊₂C₂, C₂²≀C₂.₁D₅, C₂²⋊C₄⋊₁Dic₅, C₂³⋊Dic₅⋊₂C₂, (D₄×C₁₀).₅C₂², (C₂²×C₁₀).₁₄D₄, C₂³.₅(C₅⋊D₄), C₂³.₁(C₂×Dic₅), C₁₀.₄₀(C₂³⋊C₄), C₂.₄(C₂³⋊Dic₅), C₂².₁₂(C₂³.D₅), (C₅×C₂²⋊C₄)⋊₇C₄, (C₂×C₄).₅(C₅⋊D₄), (C₅×C₂²≀C₂).₁C₂, (C₂²×C₁₀).₃₈(C₂×C₄), (C₂×C₁₀).₁₅₈(C₂²⋊C₄), SmallGroup(320,94)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₁₀ — C₂⁴⋊₂Dic₅

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂²×C₁₀ — D₄×C₁₀ — C₂³⋊Dic₅ — C₂⁴⋊₂Dic₅

Lower central

C₅ — C₁₀ — C₂×C₁₀ — C₂²×C₁₀ — C₂⁴⋊₂Dic₅

Upper central

C₁ — C₂ — C₂² — C₂×D₄ — C₂²≀C₂

Generators and relations for C₂⁴⋊₂Dic₅
G = < a,b,c,d,e,f | a²=b²=c²=d²=e¹⁰=1, f²=e⁵, ab=ba, eae^-1=ac=ca, ad=da, faf^-1=abcd, bc=cb, ebe^-1=bd=db, fbf^-1=bcd, fcf^-1=cd=dc, ce=ec, de=ed, df=fd, fef^-1=e^-1 >

Subgroups: 350 in 94 conjugacy classes, 23 normal (all characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₁₀, C₁₀, C₂²⋊C₄, C₂²⋊C₄, M₄(2), C₂×D₄, C₂×D₄, C₂⁴, Dic₅, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂³⋊C₄, C₄.D₄, C₂²≀C₂, C₅⋊₂C₈, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, C₂²×C₁₀, C₂≀C₄, C₄.Dic₅, C₂³.D₅, C₅×C₂²⋊C₄, C₅×C₂²⋊C₄, D₄×C₁₀, D₄×C₁₀, C₂³×C₁₀, C₂₀.D₄, C₂³⋊Dic₅, C₅×C₂²≀C₂, C₂⁴⋊₂Dic₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, D₅, C₂²⋊C₄, Dic₅, D₁₀, C₂³⋊C₄, C₂×Dic₅, C₅⋊D₄, C₂≀C₄, C₂³.D₅, C₂³⋊Dic₅, C₂⁴⋊₂Dic₅

Smallest permutation representation of C₂⁴⋊₂Dic₅
►On 40 points

Generators in S₄₀

(1 40)(2 36)(3 32)(4 38)(5 34)(11 37)(12 33)(13 39)(14 35)(15 31)

(1 40)(2 36)(3 32)(4 38)(5 34)(6 24)(7 30)(8 26)(9 22)(10 28)(11 37)(12 33)(13 39)(14 35)(15 31)(16 23)(17 29)(18 25)(19 21)(20 27)

(1 14)(2 15)(3 11)(4 12)(5 13)(31 36)(32 37)(33 38)(34 39)(35 40)

(1 14)(2 15)(3 11)(4 12)(5 13)(6 17)(7 18)(8 19)(9 20)(10 16)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)

(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40)

(1 16)(2 20)(3 19)(4 18)(5 17)(6 13)(7 12)(8 11)(9 15)(10 14)(21 37 26 32)(22 36 27 31)(23 35 28 40)(24 34 29 39)(25 33 30 38)

G:=sub<Sym(40)| (1,40)(2,36)(3,32)(4,38)(5,34)(11,37)(12,33)(13,39)(14,35)(15,31), (1,40)(2,36)(3,32)(4,38)(5,34)(6,24)(7,30)(8,26)(9,22)(10,28)(11,37)(12,33)(13,39)(14,35)(15,31)(16,23)(17,29)(18,25)(19,21)(20,27), (1,14)(2,15)(3,11)(4,12)(5,13)(31,36)(32,37)(33,38)(34,39)(35,40), (1,14)(2,15)(3,11)(4,12)(5,13)(6,17)(7,18)(8,19)(9,20)(10,16)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40), (1,16)(2,20)(3,19)(4,18)(5,17)(6,13)(7,12)(8,11)(9,15)(10,14)(21,37,26,32)(22,36,27,31)(23,35,28,40)(24,34,29,39)(25,33,30,38)>;

G:=Group( (1,40)(2,36)(3,32)(4,38)(5,34)(11,37)(12,33)(13,39)(14,35)(15,31), (1,40)(2,36)(3,32)(4,38)(5,34)(6,24)(7,30)(8,26)(9,22)(10,28)(11,37)(12,33)(13,39)(14,35)(15,31)(16,23)(17,29)(18,25)(19,21)(20,27), (1,14)(2,15)(3,11)(4,12)(5,13)(31,36)(32,37)(33,38)(34,39)(35,40), (1,14)(2,15)(3,11)(4,12)(5,13)(6,17)(7,18)(8,19)(9,20)(10,16)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40), (1,16)(2,20)(3,19)(4,18)(5,17)(6,13)(7,12)(8,11)(9,15)(10,14)(21,37,26,32)(22,36,27,31)(23,35,28,40)(24,34,29,39)(25,33,30,38) );

G=PermutationGroup([[(1,40),(2,36),(3,32),(4,38),(5,34),(11,37),(12,33),(13,39),(14,35),(15,31)], [(1,40),(2,36),(3,32),(4,38),(5,34),(6,24),(7,30),(8,26),(9,22),(10,28),(11,37),(12,33),(13,39),(14,35),(15,31),(16,23),(17,29),(18,25),(19,21),(20,27)], [(1,14),(2,15),(3,11),(4,12),(5,13),(31,36),(32,37),(33,38),(34,39),(35,40)], [(1,14),(2,15),(3,11),(4,12),(5,13),(6,17),(7,18),(8,19),(9,20),(10,16),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40)], [(1,16),(2,20),(3,19),(4,18),(5,17),(6,13),(7,12),(8,11),(9,15),(10,14),(21,37,26,32),(22,36,27,31),(23,35,28,40),(24,34,29,39),(25,33,30,38)]])

41 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	5A	5B	8A	8B	10A	···	10F	10G	···	10R	10S	10T	20A	···	20F
order	1	2	2	2	2	2	2	4	4	4	4	5	5	8	8	10	···	10	10	···	10	10	10	20	···	20
size	1	1	2	4	4	4	4	4	8	40	40	2	2	40	40	2	···	2	4	···	4	8	8	8	···	8

41 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+			+	+	+	-	+	-			+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	D₄	D₄	D₅	Dic₅	D₁₀	Dic₅	C₅⋊D₄	C₅⋊D₄	C₂³⋊C₄	C₂≀C₄	C₂³⋊Dic₅	C₂⁴⋊₂Dic₅
kernel	C₂⁴⋊₂Dic₅	C₂₀.D₄	C₂³⋊Dic₅	C₅×C₂²≀C₂	C₅×C₂²⋊C₄	C₂³×C₁₀	C₂×C₂₀	C₂²×C₁₀	C₂²≀C₂	C₂²⋊C₄	C₂×D₄	C₂⁴	C₂×C₄	C₂³	C₁₀	C₅	C₂	C₁
# reps	1	1	1	1	2	2	1	1	2	2	2	2	4	4	1	2	4	8

Matrix representation of C₂⁴⋊₂Dic₅ ►in GL₆(𝔽₄₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	40	39
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	40	39	0	0
0	0	0	1	0	0
0	0	0	0	40	39
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	40	0
0	0	0	0	0	40

1	0	0	0	0	0
0	1	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	40	0
0	0	0	0	0	40

19	40	0	0	0	0
25	16	0	0	0	0
0	0	1	0	0	0
0	0	40	40	0	0
0	0	0	0	1	0
0	0	0	0	40	40

7	32	0	0	0	0
1	34	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	40	40	0	0

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,39,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,39,1,0,0,0,0,0,0,40,0,0,0,0,0,39,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[19,25,0,0,0,0,40,16,0,0,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[7,1,0,0,0,0,32,34,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₂⁴⋊₂Dic₅ in GAP, Magma, Sage, TeX

C_2^4\rtimes_2{\rm Dic}_5

% in TeX

G:=Group("C2^4:2Dic5");

// GroupNames label

G:=SmallGroup(320,94);

// by ID

G=gap.SmallGroup(320,94);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,675,297,1684,12550]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^10=1,f^2=e^5,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*b*c*d,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f^-1=b*c*d,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;

// generators/relations

G = C24⋊2Dic5 order 320 = 26·5

1st semidirect product of C24 and Dic5 acting via Dic5/C5=C4

G = C₂⁴⋊₂Dic₅ order 320 = 2⁶·5

1^st semidirect product of C₂⁴ and Dic₅ acting via Dic₅/C₅=C₄